Homogeneous Stochastic Processes

Pacific Journal of Mathematics

HOMOGENEOUS STOCHASTIC PROCESSES

JOHN W. WOLL

Vol. 9, No. 1 May 1959 HOMOGENEOUS STOCHASTIC PROCESSES*

JOHN W. WOLL, JR.

Summary. The form of a stationary translation-invariant Markov process on the real line has been known for some time, and these processes have been variously characterized as infinitely divisible or infinitely decomposable. The purpose of this paper is to study a natural gene- ralization of these processes on a homogeneous space (X, G). Aside from the lack of structure inherent in the very generality of the spaces (X, G), the basic obstacles to be surmounted stem from the presence of non trivial compact subgroups in G and the non commutativity of G, which precludes the use of an extended Fourier analysis of characteristic functions, a tool which played a dominant role in the classical studies. Even in the general situation there is a striking similarity between homogeneous processes and their counterparts on the real line. A homogeneous process is a process in the terminology of Feller [3] on a locally compact Hausdorff space X, whose transition probabilities P(t, x, dy) are invariant under the action of elements g e G of a tran- sitive group of homeomorphisms of X, in the sense that P(t, g[x~}> g[dyj) = P(t, x, dy). It is shown that if every compact subset of X is separable or G is commutative the family of measures t~τP(t, x, •) converges to a not necessarily bounded Borel measure Qx( ) on X-{x} as £->0, meaning that for every bounded continuous, complex valued function / on X which vanishes in a neighborhood of x and is constant at infinity t-Ψ(t,x,f)-+Qx(f). In 3 we show that the paths of a separable homogeneous process are bounded on every bounded ^-interval and have right and left limits at every t with probability one. If the action of G on X is used to translate the origin of each jump to x, it is shown for suitably regular compact sets D that the probability of a jump into D while t e [0, T] is given by 1-exp {-TQX(D)}. The maps/->P(£, .,/) = (Γ,/)( ) map the Banach space, C(X), of continuous functions generated by the con- stants and functions with compact support into itself, and by a suitable normalization can be assumed strongly continuous for t > 0. Indeed,

Tt is a strongly continuous semi-group. The domain D(A) of the infinitesimal generator A of Tt admits a smoothing operation whose precise

* This paper was originally accepted by the Trans. Amer. Math. Soc; received by the editors of the Trans. Amer. Math. Soc. November 5, 1956, in revised form February 24, 1958. This paper is a thesis, submitted in partial fulfillment of the requirements for the Ph. D. degree, Princeton University, June 1956. During its preparation the author was a National Science Foundation Predoctoral Fellow.

293 294 JOHN W. WOLL, Jr. nature is described in Corollary 1 of Theorem 2.2. Roughly speaking, if g 6 D(A), ε > 0, and / e C(X) we can find an h e D(A) such that \\h — /|Ioo < ε and /= h on any preassigned compact subset of interior ({x\f(x) = g(x)}). A family of measures P(x, A) which is a probability measure on the Borel subsets of X for fixed x e X, measurable in x for fixed A, and invariant under G(i.e. P(g[x], g[A]) = P(x, A)), is called a homogeneous transition probability of norm one. Such a family generates a continuous endomorphism /-> P( ,/) = (P/)( ) of C(X), and a homogeneous process Tt = exp{ίr(P— 1)}. This latter process is called a compound Poisson process. In 4 we study strong convergence for compound Poisson processes (Definition 4.1) and prove among other facts that every homogeneous process is a strong limit of compound Poisson processes exp {tr^Pi — 1)}, and if rt < M < + oo the limit process is necessarily compound Poisson. If X is given the discrete topology every homogeneous process on X is compound Poisson. In case the Qx associated with P(t,x, dy) vanishes identically, or equivalently P(t, x, dy) has continuous paths, we show in 5 that P(t, x, dy) is the strong limit of compound Poisson processes whose Pι{x, dz) have support arbitrarily closed to x. In 7 we study subordination of homogeneous processes as defined by Bochner [1]. By phrasing the definition in terms of a probabilistically run clock it is shown that many processes are maximal in the partial order induced by subordination. If we follow the notation in (7.2) where exp {tS(b, A, F, z)} denotes the family of characteristic functions of a homogeneous process X(t, ω) on Euclidean w-space, we obtain the following type of result. When support (F) is compact, X(t, ω) is not subordinate to any process but itself unless support (F) is also contained in the half line R+b and A = 0. In this latter case X(t, ω) is subordinate to the Bernoulli process Z(t, ω) = tb. Actually somewhat stronger statements can be made but they are proven only for the real line. In our notation we have not distinguished between the application of a measure μ to a function / and the measure of a measurable set E, denoting these respectively by μ(f) and μ{E). The set X-Έ is denoted by Ec and the usual convention is adopted in letting C, R, R+,Z, and Z+ represent respectively the complex numbers, the reals, the non- negative reals, the integers, and the non-negative integers. I would like to take this opportunity to express my gratitude to Professor Bochner who has patiently encouraged this work, and whose own ideas are at the base of § 7.

1. Introduction. Let G be a Hausdorff, locally compact topological group and H one of its compact subgroups, We consider G as a group HOMOGENEOUS STOCHASTIC PROCESSES 295 of homeomorphisms of the left cosets of G modulo H, which we denote by G/H, the left coset xH being mapped by a e G into the left coset axH. Moreover, if we are primarily interested in the space G/H and the action of G on this space, there is no reason why the subgroup H should play a dominant role, for the homeomorphism x -> xb of G carries the left coset xH of H into the left coset xbb^Hb of b^Hb. As far as left multiplication by elements of G is concerned this is an operator homeomorphism and G/ϋ* is equivalent to Gjb~ιHb. For this reason we use the neutral letter X for the space GjH and denote the operation of a e G on x e X by a[x]. We call the system (X, G) a homogeneous space and note that X is naturally homeomorphic to any coset space of the form G\GZ where Gz = {a e G\a[z] = z). For the sake of exposition let N(X) be the Banach space of regular bounded complex Borel measures on X; CC(X) the linear space of continuous complex valued functions with compact support; CΌo(X) the closure of CC(X) in the uniform norm C(X) the Banach space of functions generated by CΌo(X) and the constant functions with the uniform norm. We use the current notation of W. Feller and denote linear transformations of N(X) by postmultiplication. In this notation a linear transformation T of CΌo(X) is denoted by the same letter as its adjoint transformation on N(X), viz. μT(f) = μ{Tf). By the expression μ > 0, μ e N(X), we mean μ is a real valued non-negative measure, and by the transformation La we refer to the isometries of CC(X), CΌo(X), C(X) and N(X) generated by translation of X by a e G, viz. (Laf)(x) = 1 x f(a- [x~i)f μLa(E) = μ(a[E]). When x e X we denote by δ the measure placing a unit mass at x, so that δx(E) = 1 if x e E and 0 otherwise. x aίxl For example, we shall often use the relationship S La-i= δ . Finally, when we say that a directed sequence (μq)qeQ of measures-commonly called a net in N(X) - converges weakly to μ e N(X), in symbols μQ-+ μ, we mean for every / e CC(X), μq(f) -> μ(f) as complex numbers.

DEFINITION 1.1. A homogeneous transition probability is a continuous endomorphism P: N(X)-+ N(X) satisfying: (i) μ > 0 implies μP > 0

(ii) μq-*μ implies μqP-+μP;

(iii) PLa = LaP. An endomorphism, P, with properties (i) and (ii) is usually called a transition probability and (iii) makes the transition probability homogeneous. When fe C^X) it follows from (ii) that SxP(f) is continuous in x. In addition δxP(f) e CΌo(X). To show this let αjz] be any directed sequence in X tending to infinity, then by virtue of the assumed compactness of Gzy a^D] -> infinity for any compact set D c X and 296 JOHN W. WOLL, Jr.

La-ιf(y) = f(

= δ*LarlP(f) = δΨ(Larlf) -+ 0 ,

proving δ*P(f) e C^X). Now for any given μ e N(X) let μq = Σ ί^V^ be a bounded, directed sequence of purely atomic measures in N(X) ap- proaching μ weakly, μq -> μ. Then separate calculations show that on the one hand μQP(f) -> μP(f), while on the other

μJP(f) = Σim^ Accordingly, for / e C4-X") and μ 6 N(X)

(1.1) /<(«*[/]) - μP(f) the adjoint of P transforms CΌo(X) into itself and when / e CJ^X)

(1.2) (Pf)(x) = By letting / f 1 in (1.1) we see that (1.1) and (1.2) hold for fe C(X) as well. If P and Q are homogeneous transition probabilities

(1.3) (1.4) IIPQII = I|P|| HQIl. We obtain (1.3) by noting first from (iii) that <5*P(1) does not depend on x G X, and then it follows from (i) that μ > 0 implies \\μP\\ = μP(l) — II^H^P(l); so ||P 1| > ^XP(1). The opposite inequality is obtained by using the preceeding remarks on a Jordan decomposition of μ e N(X), while (1.4) follows easily from (1.3). We remark at this point that if z e X and m is the normalized

Haar measure of Gz, then the mapP->Pe N(G), given for geC^G) by P(g) = 1 δ'P(dy)\ m(dw)g(yw), maps the Banach algebra generated by the homogeneous transition probabilities isometrically onto the subalgebra m * N(G) * m of N(G).

DEFINITION 1.2. A homogeneous process is a one-parameter semi- group, (T£)ί>0, of homogeneous transition probabilities which is temporally continuous in the sense

(iv) μ e N(X) and / e C(X) imply μTt(f) is continuous in t.

vt Using (1.4), (1.3) and (iv) we see that ||Γt|| = e . Therefore, we pt replace Tt by the equivalent process e~ Tt and assume in the rest of this paper that (v) below is satisfied unless explicitely stated otherwise.

(v) \\Tt\\ = 1, HOMOGENEOUS STOCHASTIC PROCESSES 297

The requirement of temporal continuity for a homogeneous process is equivalent to the weak continuity of the restricted adjoint process

Tt: C(X) -• C(X), and by well known results, [5], implies the strong continuity of this last process for t > 0. The theory of semi-groups shows that Tt is strongly continuous at t — 0 on the closure of

\Jt>0TtC(X). This may be a proper subspace of C(X). For example, let (G, G) be the homogeneous space of a group acting on itself by left translations with μTt = μ * m, where m is the normalized Haar measure of a non trivial compact subgroup. G. Hunt [7, pp. 291-293] has shown that by enlarging the subgroups Gz if necessary we can always assume a homogeneous process possesses the following quality.

Property I. For any fixed x e X and any Borel measurable neigh- x x c borhood N of x, d Tt(N) -> 1 or equivalents δ Tt(N ) -> 0 as t -> 0. It is a routine calculation to show that Property I is equivalent to the strong continuity of Tt: C(X) -> C(X) at the origin, so we hence- forth assume our processes satisfy Property I, and by Hunt's result we can do this without loss of generality. In view of the above statements we can apply the Hille-Yosida theory of strongly continuous semi-groups to the semi-group Tt: C(X) ~> C(X). An elementary application of this theory shows that there exists a dense linear subspace of C(X) which we denote by D(A), and a closed linear operator A: D(A) -> C(X) with the property that for / e D(A) - A/ΊU = 0.

2. Properties of D(A). In this section we investigate the domain of the infinitesimal generator for homogeneous processes which satisfy Property II below. Later we show that Property II is automatically satisfied if either X is separable or G is commutative.

Property II. There is a regular Borel measure Qz on X — {z}, such that t-\TJ){z) -> Qβ(f) as t -> 0 for all / e C(X) which vanish on any neighborhood of z.

Qz is positive and QzLa-λ = Qα[β]. In general, of course, Qz will be unbounded, although its values on any set E lying in the complement of a fixed neighborhood N of z must be bounded, or equivalently

δ*Tt(E) = O(|ί I) as t -H>0. This is easily checked if one notes that D(A) includes the constant functions and is invariant under G. For then we can choose an / 6 D{A) which is everywhere positive on X, vanishes at z, and is greater than 1 on Nc. Clearly

QZ(E) < (Af)(z) = limt^t'\Ttf)(z) is bounded independently of E c NG.

The homogeneity of the process Tt entails a uniformity in this 298 JOHN W. WOLL, Jr.

convergence to Qz which may be stated as follows.

THEOREM 2.1. // / is an open subset of X, fe C(X), and f(J) — ι 0 then t~Ttf converges to its limit as t -> 0 uniformly on every compact subset of J. If in addition Jc is compact, this convergence is even uniform on every closed subset of J.

Proof. To prove the first assertion it suffices to show that the approach is uniform on a neighborhood of z e J of the form N[z] where N2[z] c J and N = N'1 is compact. If α, b e N, and K is chosen so e that t-ψTt(N[z] ) < K, then

|t-i3αωΓί(/) _ t-^Tt{f)\< K\\LhaM) ~ f\\-

ι x The family {ht(x)\t > 0, ht(x) = t-δ Tt(f)} is accordingly, equicontinuous

on N\z\, and ht(x) -> Qx(f) as t -* 0. It follows that this approach is uniform on N[z], The second assertion is a consequence of the fact

that for each e > 0 there is a compact set Ds and a ts > 0, such that

x $ D2 and t < tε imply \t'\Ttf)(x)\ < ε. Assume on the contrary that

there is a sequence at[z] -> infinity, together with a sequence ί, -• 0, 1 such that \tr (Ttif)(ai[z'])\ > s. Now choose a bounded sequence of functions (^ from C(X) which converges to zero monotonely while their supports approach infinity, and which satisfy the crucial inequalities

g3 > sup^j|Z/αrl(/)|. The inequalities

1 ε < liminf<|ti- (Γί4/)(α,M)| < Km^fo) = 0 yield the contradiction which proves the second assertion. Suppose h 6 D(A) and on some open subset J of X we have / = h, where fe C{X). Suppose further that either / or Jc is compact. Let D be a closed subset of J, containing a neighborhood of infinity if Jc is compact; and let m e C(X) be constructed so that o R defined by

S(x, y, t) = (Tm(yytf)(x) is, because of the strong continuity of Tu defined and continuous in (x, y, t). Consequently its restriction to the diagonal in the first two components is continuous on X x (0, oo). When G ^^(^ΛW^tW), while if J is compact (Tmωtf)(x) =

(TJ)(x) = f{x) close to infinity. The maps Wt:C(X) -> C(X) defined by

fix) -> Tm(x)tf(x) form a strongly continuous family of bounded linear transformations, so that we may form the Rieman integral gix) =

r~\ (Wtf)(x)dt. If x 6 D, m(x) = 0 and g(x) = fix); while for any x, Jo

1 P \g(x) - f(x)\ = r- 1 f {TΓt/(a5) - f(x)}dt\ JO

< supί 0 as r -»- 0 . HOMOGENEOUS STOCHASTIC PROCESSES 299

Thus for any given ε > 0 we may guarantee that \\g — /|U < ε by a sufficiently small choice of r.

THEOREM 2.2. Let D be a closed subset of the open set J in X, where either J or Jc is compact. Let h e D(A), f e C(X), and f = h on J.

Then for each ε > 0 there exists a g,: e D(A)y such that (i) h = / = gs on

D, and (ii) ||#ε-/|L<ε.

Proof. We only need to prove that the g defined above lies in D(A). To do this we show that

\- f}dt converges to its limit uniformly on X. The computations are divided into two cases.

Case I. m(x) < 1/2. On this closed subset of J

1 -h} +s- {Ts(f~h)~(f-h)} .

x Now s~ {Tsh — h} -> Ah in the uniform norm as s -• 0, and by Theorem

2.1 s-'iT^f- h) - (/- h)} ->Qx(f- h) uniformly on {x\m(x) < 1/2}. Accordingly,

τ 1 s' {Tsg - g} ^r- Γ Wt{Ah + Qx(f - h)}dt Jo uniformly on {x\m(z) < 1/2).

Case II. m(x) > 1/4. On this set

1 s-^Tsg - g} = (rs)' [ {Ts+mix)tf - Tm(x)tf}dt Jo [ Γ m(x}r+s Γs 1 {T f}dn- T fdu\ m(x)r J u Jθ u J

-+ {rnι(x)}-i{Tmωrf(x) - f(x)} as s -> 0, uniformly on {x\m(x) > 1/4}. These observations show that g 6 D{A). In the above proof we did not really need the fact that h e D(A). Indeed, had we replaced h by h + u, where u e C(X) and u{J) — 0, there would have been no change in the proof. Furthermore, since ι g(x) = r~ \ (Tm{x)tf){x)dty if / is real valued we have the relation Jo infx f(x) < inf yg(y) < sup^(2;) < supx/(x). These remarks allow us to state a corollary. 300 JOHN W. WOLL, Jr.

COROLLARY 1. Let Jlf J2, •••, Jn be disjoint open sets of Xand let

Dt be a closed subset of Jt. Suppose that for each i either Jt is compact or for at most one i, J\ is compact. Let ht e D{A) and f e C(X) satisfy f —hi on Jt. Then for each ε > 0 there exists a g2 eD(A)> such that

(i) ge = ht = / on Di9

(ϋ) |lflrβ-/||-

(iii) iff is real valued then mfxf(x)< mf yg£(y) < sup* gs(z)

COROLLARY 2. If B is compact in X, His closed, and H Π B — φ; there exists an f e D(A), such that 0 < / < 1, f(H) = 0, and f(B) = 1. The above results have been derived under Property II and we now show that we can replace Property II by a condition on D(A).

Property III. For each ε > 0 and / e C(X) which is zero in a neighborhood J of x9 there exists an /ε e D(A), such that fs = 0 in a neighborhood U of x which is independent of ε, ||/ε — /|U < ε, and if / is real valued so is fe with inΐxf(x) < infyfB(y) < supz f,(z) < supxf(x).

THEOREM 2.3. Property III is equivalent to Property II for homogeneous processes.

Proof. We have already seen that Property II implies Property III and will now demonstrate the converse. First, let / be a neighborhood of x, and choose a function h e C(X), such that h = 0 in a neighborhood U d J of x, and h(JG) — 1. We also require that h > 0 everywhere.

Now for 0 < ε < 1 choose an hs in accordance with Property III.

λ G 1 t- d*Tt{J ) < (1 - eyWT^t- < K{U)< + oo for some constant K(U) depending on U. If /e C(X) and /(J) = 0, 1 x Property II is equivalent to the fact that t~ d Tt{f) -> a limit as t -> 0. The inequalities

) - \im t-WTt{fH)\ <

< (ε, + e,)lim sup t-WTW) < (ε, + et)K(ϋ) show that lim lim ί'^Γ^/g) exists. If we call this limit 6, the inequality

\WTt(f) -b\< \t-WTt(fe) - 6|+ εK{U) HOMOGENEOUS STOCHASTIC PROCESSES 301 shows by first letting t -> 0 and then ε -> 0, that t"\TJ){x) ->b as t -> 0, completing the proof. By the use of well-known smoothing techniques which exist on a differentiable manifold we can conclude that if G is a Lie group or if X is a differentiable manifold on which G acts differentiably, and D(A) includes C°°(X) Π C(X), then Property III and Property II are automatically satisfied. We close this section by remarking that the only place where we have used the homogeneity of Tt was in the proof of Theorem 2.1, so that a strongly continuous semi-group on C(X) satisfying the conclusions of Theorem 2.1 also satisfies the conclusions of Corollary 1 if its adjoint preserves positivity.

3* The paths of a homogeneous process and Property IL Before we discuss the nature of the paths of a homogeneous process it is necessary to take certain precautions which will assure us that the properties we want to discuss can be handled by the theory of probability. Given a consistent set of transition probabilities for a particle moving in a locally compact Hausdorff space it is possible, using a theorem of Kolmogoroff s, to construct an infinite product space in which these transition probabilities determine the finite dimensional distributions. From this we can construct, in the usual manner, a set of paths and a probability measure on this path space. Alternatively, we can consider a process as a family X(t, ω) of measurable transformations from some abstract sample space Ω to the range space X. Since there is some freedom in the definition of X(t, ω) given only the finite dimensional distributions, it is important to notice that the concept of separability as used on the real line is available in this case.

DEFINITION 3.1. A process {X(t, ω), 0 < t) will be called separable relative to the class A of closed subsets of the locally compact Hausdorff space X if, and only if, (1) there exists a denumerable subset {tj} c [0, oo), and (2) an event K c Ω with P{K} — 0, such that for every open interval Ic. [0, oo), and every set F e A, the event

{ω\X(tjf ω) β F, tj 6 I Π {tj}} - {ω\X(t, ω) 6 F, t € 1} C K .

The importance of the concept of separability rests on the validity of the following theorem.

THEOREM 3.1. Let Xbe a separable locally compact Hausdorff space, and let {X(t, ω), t < 0} be an X-valued process. Then there exists an X*-valued stochastic process {X(t, ω), t > 0} such that: (1) X{t, ω) is 302 JOHN W. WOLL, Jr. defined on the same ω-space, Ω, as X(t, ω) and takes values in the one point compactification, X*, of X; (2) X(t, ω) is separable relative to the class of closed sets of X* (3) for every t>0 {Pω\X(t, ω) = X(t, ω)} = 1.

Proof. This theorem may be proved in a manner entirely analogous to the comparable theorem on the real line, and its proof is given, for example, in Doob [2 p. 57]. We remark that the separability of the space is necessary for this proof. In the following we discuss the displacements of the path X(t, ω) during a closed interval of time. The success of our technique depends directly on the possibility of comparing two such displacements with different origins. If our homogeneous space (X, G) is that of a group acting on itself, we can translate all displacements origins to the identity and their endpoints are uniquely determined. On a general homogeneous space, however, the endpoint of a translated displacement is not determined by its new origin. We introduce several concepts from the calculus of relations and a space of displacements to handle this ambiguity. By a relation (U) on X we mean a subset of X x X which contains the diagonal. The notations (U)-1 = {(y, x)\(x, y) e (U)}, (W) o (U) = {(x, y)\ for some z, (x, z)e(U) and (z, y) e(W)}, and (U)[A]= {x\(y, x)e(U) for some y e A} are standard. It is sometimes convenient to substitute

Ux for (U)[{x}] and in this notation (W) o (U)x = \Jveϋx Wy. A relation (U) is called homogeneous whenever (x, y) e (U) implies (α[x], a[yj) e (U) for all a e G, and it is convenient to describe a relation by giving a property possessed by all the sets Ux. For example, we call a relation

(U) compact, open, closed, or a neighborhood relation if each Ux is compact, open, closed or a neighborhood of x respectively. Using a double coset representation it is easily shown that the class of homogeneous neighborhood relations forms a base for the natural uniformity of the homogeneous space (X, G). We say the displacement from x to y belongs to the homogeneous relation (£7) if (x,y) e (U). The reason for this mention of relations is that they appear to be exactly what is needed to generalize the statement and proof of a theorem from Kinney [9], p. 292-293.

THEOREM 3.2. Let (X, G) be a homogeneous space satisfying the first axiom of countability, and let X* be the space X compactified by adding a single point at infinity. Let {X(t, ω), t > 0} be an X*-valued homogeneous stochastic process governed by the transition probabilities x δ Tt. and separable with respect to the closed subsets of X*. Then if

T > 0, there is an ω-set Eτ with P{ET) = 0, such that ω 0 Eτ implies the statements below. (1) X(t, ώ) is bounded on t e [0, T). By which we mean HOMOGENEOUS STOCHASTIC PROCESSES 303

Cl[{X(t, ω)\t β [0, T)}] is a compact subset of X. (2) X{t, ω) has finite right and left hand limits at every t e [0, T). (3) The number of jumps in [0, T) whose displacements lie out- side a homogeneous neighborhood relation (U) is finite. Furthermore, for any homogeneous neighborhood relation, (U), the maximum number of disjoint subintervals (t,s) o/[0, T) for which (X(t~,ω), X(s~,ω)) $ (U) is finite, where X(t~, ω) = \imh]QX(t + h, ω). In particular the use of the one point compactification of X was only necessary to cover the processes constructed in Theorem 3.1 and may be eliminated as soon as (1) is proved. The displacement or jump of a particle from x to y can be considered as a point in the space X x X. It is natural to consider classes of similar displacements, and this involves the introduction of an equivalence relation on X x X, two points, (x, y) and (#', yf), being considered equivalent if there is an a e G such that (a[x~], a[yj) = (xf, yf). This is a closed equivalence relation and the quotient space is homeomorphic to the space of double cosets {GxaGx} = Y. Let p:lxl->7be the canonical projection. Y is a locally compact Hausdorff space known as the space of displacements, and p is a continuous open mapping. If we fix the first component at x, so we only consider jumps origination at xf we get a mapp':X-> Y given by z-+p(x,z). Using this map the commutativity of Tt with La shows that we can very properly place x the measures δ Tt and Qx on the space Y without losing a thing. If

/ e CC(X), and m is the normalized Harr measure of Gx, the equation x x x δ Tt(f) = [f(a[z})m(da)δ Tt(dz) indicates a means of returning 3 Tt and

Qx to X from Y. The following theorem is the key to the results of this section. It and Theorem 3.4 are generalizations of similar results for homogeneous processes on the real line which may be found, for example, in Doob [2, p. 422-424].

THEOREM 3.3. Let (X, G) be a homogeneous space satisfying the first axiom of countability, and X{t, ώ) a separable homogeneous process on Σ X governed by the transition probabilities δ Tt{ ). Suppose that for some sequence of t-values, t5 -> 0, there is a not necessarily bounded regular Borel measure Qx on X — {x} for which f e CC(X), x 0 sup- 1 port (/), imply t] (Ttif)(x)-+Qx(f) as t3 -> 0. Let Y be the space of x displacements of (X, G), and denote the measures δ Tt and Qx after transference to Y by the same symbols. Suppose X(0, ω) — x a.s,, C(D) = {ωllim^ooPίXίί - n~\ ω), X(t + n'\ ω)) e D for some t e [0, I7)}, and let P* and P* be the inner and outer measures induced by P on Ω. (1) Then for any compact subset D of Y — {x}: 304 JOHN W. WOLL, Jr.

1 x 1 - exp {T\imsuv3tj δ Tt(D)} < P*{C(Z>)} P (2) If D is a compact subset of Y — {x} satisfying, either x (i) there is a Uopen c D for which δ Tt(U — D) = o(t) as ί -• 0, or (ii) £Λere exists a sequence C^ compact c interior (D) for which

Qx(Ck) -> QX(D) as k -> oo, = P*{C(Z>)} = 1 - e-Γβ«w

Proof. Using the monotone sequence t3 -> 0 we define a sequence of partitions of [0, Γ). The ith partition being given by

{[0, tj), ltJf 2tj), , [(kj - l)tj, kjtj), [kjtj, T)} , where k3 is the largest integer < Tjt3. Define Y-valued random variables

= p(X({n - l}tj'f ω), X{ntf, ω)) l

+ l) = p(X(kjtf9 ω)X(T-, ω)) , where as usual X(t~, ω) — limsTcX(s, ω). For any measurable subset FdYwe put F(j, n) = {ω\H(j, n) eF} and F(j) = (JV1 F(j, n). Since

{H(j, ri)yl

P{F(j)} = 1 - {1 - δ*Tτ_kjtj(F)}{l - δ*Ttj(F)}*> ,

1 where 1 > ε, -> 0, 0 < T - kόt3 < t5 \ 0, and k5 = Ttf - ε3. Let D be a compact subset of Y, and Z7 an open neighborhood of D whose closure does not contain x. Our knowledge that the paths have right and left hand limits at every point shows that C(D) c lim inf3 U(j). If

ω e lim sup^ D(j), choose a sequence of semiclosed intervals [n3t3, (n3 + l)t3) which converge to a point, and across whose length the path ω has a displacement H(j, n3) e D. By passing to further subsequences if necessary we can assume that the sequences of endpoints are monotone. There are four cases;

(a) njtj f, (nj + ΐ)t31 leads to ω e C{D)

(b) itjtj I, (n3 + l)tj t is impossible; while

(d) n3t31, (n3 + l)t31 both lead to ω's having infinitely many displacements close to D in [0, Γ), and, accordingly, occuring with probability zero by condition (3) of Theorem 3.2. Therefore, lim sup^ D(j) c

C(D) c lim inf 3 U(j) which in turn implies

1 _ exp {- Tlimsup^r^CD)} < and P*{C(D)} < 1 - exp {- HOMOGENEOUS STOCHASTIC PROCESSES 305

It can be shown by a standard arguement that if V open 3 U and x 0 V, ι x then QX(D) < lim inf} tfd Tt){U) < QX{V), and by letting Fand Ushrink to .D it follows that P*{C(D)} < 1 - e-^^) . If in addition D satisfies either (i) or (ii) in (2), it is easy to see that which implies the conclusion of (2). We now show that the conditions (i) and (ii) placed on the compact set D in (2) of Theorem 3.3 are sufficiently unrestrictive for us to prove Property II for homogeneous processes.

THEOREM 3.4. Let (X, G) be a separable locally compact homogeneous space, and let Tt be a homogenous process on (X, G). Then there exists a unique not necessarily bounded Borel measure Qx on Y — {x}, the space of displacements of (X, G)> such that f e C(X) and x $ support (/) imply

t'\Ttf){x) -> Qx(f) as £->0. Proof. The use of the Hille-Yosida theory of strongly continuous semi-groups shows that when restricted to the complement of any neighbor- 1 x hood U of x e Y, the family of measures t~ δ Tt is bounded. We compact- if y Y by adding a point at infinity and denote the compactified space by Y*. Using the compactness of bounded sets in the weak star topology, and the first axiom of countability for Y, we can find a sequence tό -> 0 and a not necessarily bounded Borel measure Qx on F* — {x}, such that ι x for any feC(Y) = C{ Γ*), x φ support (/) implies tfd Ttμ) as t3 -> 0. There remain two problems.

(a) to prove Qx is unique, and

(b) to show that Qx({^}) = 0. Using the separability of (X, G) construct a representation, X(t, ω), of the paths of Tt satisfying all the conditions in Theorem 3.2. 1 x Suppose that sf S TSj -> QJ were another limit and let C be any

compact subset of 7- {x}. For an arbitrary ε > 0 let Us be chosen

so that C a Uζ open c Us compact c Y- {x}, QX(U,) < QX(C) + ε, and

QX'(US)< QX'(C) + ε. Now construct a function f e Ce(Y), such that f(U,c) = 0, 0 < / < 1, and f(C) = 1. Select a 6 e (0,1) such that

\f() - ί>}) - o and put D = {x\f(x) < b}. Note that

C compact c interior (D) c D compact c UC9 306 JOHN W. WOLL, Jr.

Qx (interior (D)) = QX(D), and QJ (interior (D)) = QJ(D). Accordingly, D satisfies condition (ii) of Theorem 3.3 (2), and thus for any fixed T > 0,

τ rρ P*{C(D)} = 1 - e~ Qx(D) = 1 - e- *'W ,

so that QX(D) = QJ{D). This shows

\QΛC) ~ QX(C)\ < \QX(D) - QX(D -C) + QΛD - C) - QX'(D)\ < 2ε

r for every ε > 0; so Qx = Qx .

To show that Qx is really a Borel measure on F- {x}, and not on

F* — {x}9 we must show that Qx({oo}) = 0. In order to do this we make a final appeal to the paths. Except for an event of probability zero we know that Cl[{X(t, ω)\t e [0, T)}] is a compact subset of X, and, consequently, its projection on Y will also be compact. Using the method above, choose a sequence of compact sets Dn[ {cχ>} in F* and satisfying condition (ii) of Theorem 3.3 (2). Then if Qx({oo}) ψ 0,

P{C(D)} =l-e-τQx(Dn)>r>0, and P{f\ n-\ C{Dn)} > 0. Any path with a jump in every Dn during the time interval [0, T) certainly contains co as a limit point. Thus our process violates the condition that the paths are a.s. bounded. Hence Q,({«>}) = 0.

The temporal continuity (weak continuity) of Tt enables us to restrict x consideration to a sigma-compact subset of X, namely U t>0 support (δ Tt). As a consequence of the preceeding theorem this remark proves the following corollary.

COROLLARY. Let (X, G) be a homogeneous space where every compact subset of X is separable. Then every homogeneous process on (X, G) possesses Property II. The following theorem gives an accurate description of the important set support (Qx).

x THEOREM 3.5. Let (X, G), Y, X(t, ω), δ Tt, and Qx be as in Theorem 3.4. For those ω's which have one sided limits let F{ω) = {limp(X(ί - n~\ ω), X(t + n~\ ω))\t > 0} .

Then F(ώ) = support (Qx) U {x} a.s..

Proof. By definition CflJΛ) = U«C(Fa) for FΛ measurable c Y.

Now choose a sequence of compact sets Di9 such that U

which shows F(ω)dsupport(QX)U {#}. Let Uopenc Ucompactc Y— {x}, and U Π support (Qx) =£ φ. Choose / e Cc( Γ), such that 0 < / < 1, support (/) c U, and Qx(f) > 0. Let

r(U) = lim sup t'^xTt(U> Qx(f) > 0 . £->U

If {Ui9 i = 1, 2, , n} is a finite class of such U's, let

7 P*{ω|ω has a jump in every Ut when ί < ϊ } = A(T) 1 1 P*{ω|ω has a jump in C7cf when T(k - l)^" < ί < Tkn" } = 5(fc, Γ) .

We have the following inequalities relating the above numbers

A(T) > J5(l, Γ)5(2, Γ) B(n, T)

ι > (1 - exp {- ^^ΓrίC/,)}) . (1 - exp {- n-Tτ{Un)}) . Letting T approach infinity one sees that

P^{ω\ω has a jump in each ί7έ for some t > 0} > 1. If we now choose a countable sequence of finite classes of the type {[/«} above, and let their sets become arbitrarily fine while their unions swell out and eventually cover support (Qx),

P* {ω I ω has a jump in each Z7t of the fcth covering} = 1 . Hence the inner measure of their intersection is one. Now the paths in their intersection have jumps in the closure of each of the finer and finer covering sets, and consequently for these paths Cl [F(ω)] D support (Qx). The existence of left hand limits for X(t, ω) implies x e [F(ω)].